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About 3307 results. 1294 free access solutions
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| m54256 | Given the multistep method
wi+1 = −3/2 wi + 3wi−1 - 1/2 wi−2 + 3hf (ti ,wi), for i = 2, . . . , N − 1,
with starting values w0, w1, w2:
a. Find the local truncation error.
b. Comment on consistency, stability, and convergence. |
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| m54257 | Given the partition x0 = 0, x1 = 0.05, and x2 = 0.1 of [0, 0.1], find the piecewise linear interpolating function F for f (x) = e2x. Approximate
F(x) dx, and compare the results to the actual value. |
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| m54258 | Given the partition x0 = 0, x1 = 0.05, x2 = 0.1 of [0, 0.1] and f (x) = e2x:
a. Find the cubic spline s with clamped boundary conditions that interpolates f.
b. Find an approximation for
c. Use Theorem 3.13 to estimate max0≤x≤0.1 |f (x) − s(x)| and
d. Determine the cubic spline S with natural boundary conditions, and compare S(0.02), s(0.02), and e0.04 = 1.04081077. |
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| m54310 | (i) Determine if the following matrices are positive definite, and if so, (ii) construct an orthogonal matrix Q for which Qt AQ = D, where D is a diagonal matrix.
a.
b.
c.
d. |
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| m54311 | (i) Determine if the following matrices are positive definite, and if so, (ii) construct an orthogonal matrix Q for which QtAQ = D, where D is a diagonal matrix.
a.
b.
c.
d. |
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| m54331 | (i) Use Gaussian elimination and three-digit rounding arithmetic to approximate the solutions to the following linear systems. (ii) Then use one iteration of iterative refinement to improve the approximation, and compare the approximations to the actual solutions.
a. 0.03x1 + 58.9x2 = 59.2,
5.31x1 − 6.10x2 = 47.0
Actual solution (10, 1)t .
b. 3.3330x1 + 15920x2 + 10.333x3 = 7953,
2.2220x1 + 16.710x2 + 9.6120x3 = 0.965,
−1.5611x1 + 5.1792x2 − 1.6855x3 = 2.714
Actual solution (1, 0.5,−1)t .
c. 1.19x1 + 2.11x2 − 100x3 + x4 = 1.12,
14.2x1 − 0.122x2 + 12.2x3 − x4 = 3.44,
100x2 − 99.9x3 + x4 = 2.15,
15.3x1 + 0.110x2 − 13.1x3 − x4 = 4.16
Actual solution (0.17682530, 0.01269269,−0.02065405,−1.18260870)t .
d. πx1 − ex2 + √2x3 − √3x4 = √11,
π2x1 + ex2 − e2x3 + 3/7x4 = 0,
√5x1 − √6x2 + x3 − √2x4 = π,
π3x1 + e2x2 − √7x3 + 1/9 x4 = √2.
Actual solution (0.78839378,−3.12541367, 0.16759660, 4.55700252)t . |
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| m54529 | In 1224, Leonardo of Pisa, better known as Fibonacci, answered a mathematical challenge of John of Palermo in the presence of Emperor Frederick II: find a root of the equation x3 +2x2 +10x = 20. He first showed that the equation had no rational roots and no Euclidean irrational root-that is, no root in any of the forms a±√b,√ a±√b, √(a ±√b), or √√((a) ±√b), where a and b are rational numbers. He then approximated the only real root, probably using an algebraic technique of Omar Khayyam involving the intersection of a circle and a parabola. His answer was given in the base-60 number system as
1 + 22(1/60)+ 7(1/60)2 + 42(1/60)3 + 33(1/60)4+ 4(1/60)5+ 40(1/60)6.
How accurate was his approximation? |
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| m54543 | In a circuit with impressed voltage E having resistance R, inductance L, and capacitance C in parallel, the current i satisfies the differential equation
Suppose C = 0.3 farads, R = 1.4 ohms, L = 1.7 henries, and the voltage is given by
E(t) = e−0.06πt sin(2t − π).
If i(0) = 0, find the current i for the values t = 0.1 j, where j = 0, 1, . . . , 100. |
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| m54544 | In a circuit with impressed voltage E(t) and inductance L, Kirchhoff s first law gives the relationship
E(t) = L di/dt + Ri,
Where R is the resistance in the circuit and i is the current. Suppose we measure the current for several values of t and obtain:
Where t is measured in seconds, i is in amperes, the inductance L is a constant 0.98 henries, and the resistance is 0.142 ohms. Approximate the voltage E(t) when t = 1.00, 1.01, 1.02, 1.03, and 1.04. |
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| m54559 | In a paper dealing with the efficiency of energy utilization of the larvae of the modest sphinx moth (Pachysphinx modesta), L. Schroeder [Schr1] used the following data to determine a relation between W, the live weight of the larvae in grams, and R, the oxygen consumption of the larvae in milliliters/hour. For biological reasons, it is assumed that a relationship in the form of R = bWa exists between W and R.
a. Find the logarithmic linear least squares polynomial by using
ln R = ln b + a lnW.
b. Compute the error associated with the approximation in part (a):
c. Modify the logarithmic least squares equation in part (a) by adding the quadratic term c(lnWi)2, and determine the logarithmic quadratic least squares polynomial.
d. Determine the formula for and compute the error associated with the approximation in part (c). |
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| m54560 | In a paper entitled "Population Waves," Bernadelli [Ber] (see also [Se]) hypothesizes a type of simplified beetle that has a natural life span of 3 years. The female of this species has a survival rate of 1/2 in the first year of life, has a survival rate of 1/3 from the second to third years, and gives birth to an average of six new females before expiring at the end of the third year. A matrix can be used to show the contribution an individual female beetle makes, in a probabilistic sense, to the female population of the species by letting ai j in the matrix A = [ai j] denote the contribution that a single female beetle of age j will make to the next year s female population of age i; that is,
a. The contribution that a female beetle makes to the population 2 years hence is determined from the entries of A2, of 3 years hence from A3, and so on. Construct A2 and A3, and try to make a general statement about the contribution of a female beetle to the population in n years time for any positive integral value of n.
b. Use your conclusions from part (a) to describe what will occur in future years to a population of these beetles that initially consists of 6000 female beetles in each of the three age groups.
c. Construct A−1, and describe its significance regarding the population of this species. |
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| m54589 | In an electric transmission line of length l that carries alternating current of high frequency (called a "lossless" line), the voltage V and current i are described by
∂2V / ∂x2 = LC ∂2V / ∂t2, 0< x < l, 0 < t; ∂2i / ∂x2 = LC ∂2i / ∂t2, 0< x < l, 0 < t;
where L is the inductance per unit length, and C is the capacitance per unit length. Suppose the line is 200 ft long and the constants C and L are given by
C = 0.1 farads/ft and L = 0.3 henries/ft.
Suppose the voltage and current also satisfy
V(0, t) = V(200, t) = 0, 0 < t;
V(x, 0) = 110 sin πx / 200, 0≤ x ≤ 200;
∂V / ∂t (x, 0) = 0, 0 ≤ x ≤ 200;
i(0, t) = i(200, t) = 0, 0 < t;
i(x, 0) = 5.5 cos πx / 200, 0≤ x ≤ 200;
and
∂i / ∂t (x, 0) = 0, 0 ≤ x ≤ 200.
Approximate the voltage and current at t = 0.2 and t = 0.5 using Algorithm 12.4 with h = 10 and k = 0.1. |
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| m54594 | In calculating the shape of a gravity-flow discharge chute that will minimize transit time of discharged granular particles, C. Chiarella, W. Charlton, and A.W. Roberts [CCR] solve the following equations by Newton s method:
(i)
(ii) fN (θ1, . . . , θN) = Δy ∑Ni=1 tan θi − X = 0, where
a.
b.
The constant v0 is the initial velocity of the granular material, X is the x-coordinate of the end of the chute, μ is the friction force, N is the number of chute segments, and g = 32.17ft/s2 is the gravitational constant. The variable θi is the angle of the ith chute segment from the vertical, as shown in the following figure, and vi is the particle velocity in the ith chute segment. Solve (i) and (ii) for θ = (θ1, . . . , θN)t with μ = 0, X = 2, Δy = 0.2, N = 20, and v0 = 0, where the values for vn and wn can be obtained directly from (a) and (b). Iterate until || θ(k) − θ(k−1) ||∞ < 10−2. |
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| m54595 | In calculus, we learn that e = limh→0(1 + h)1/h.
a. Determine approximations to e corresponding to h = 0.04, 0.02, and 0.01.
b. Use extrapolation on the approximations, assuming that constants K1, K2 . . . exist with
e = (1 + h)1/h + K1h + K2h2 + K3h3 + · · · , to produce an O(h3) approximation to e, where h = 0.04.
c. Do you think that the assumption in part (b) is correct? |
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| m54608 | In Example 1 the Fourier series was determined for f (x) = |x|. Use this series and the assumption that it represents f at zero to find the value of the convergent infinite series |
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| m54612 | In Example 3 it is stated that for all x we have | sin x| ≤ |x|. Use the following to verify this statement.
a. Show that for all x ≥ 0we have f (x) = x−sin x is non-decreasing, which implies that sin x ≤ x with equality only when x = 0.
b. Use the fact that the sine function is odd to reach the conclusion. |
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| m54613 | In Example 3 the eigenvalues were found for the matrix A and the conditioned matrix AH. Use these to determine the condition numbers of A and AH in the l2 norm, and compare your results to those given with the Maple commands Condition Number(A,2) and Condition Number(AH,2). |
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| m54627 | In Exercise 10 of Section 3.4 data were given describing a car traveling on a straight road. That problem asked to predict the position and speed of the car when t = 10 s. Use the following times and positions to predict the speed at each time listed. |
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| m54630 | In Exercise 17 of Section 7.3 a techniquewas outlined to prove that the Gauss-Seidel method converges when A is a positive definite matrix. Extend this method of proof to show that in this case there is also convergence for the SOR method with 0 < ω < 2. |
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| m54631 | In Exercise 26 of Section 1.1 a Maclaurin series was integrated to approximate erf(1), where erf(x) is the normal distribution error function defined by
a. Use the Maclaurin series to construct a table for erf(x) that is accurate to within 10−4 for erf (xi), where xi = 0.2i, for i = 0, 1, . . . , 5.
b. Use both linear interpolation and quadratic interpolation to obtain an approximation to erf(1 3). which approach seems most feasible? |
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